Higher-order estimates of highest waves of the Whitham equation
Abstract
The Whitham equation has given its name to a wider family of nonlocal, nonlinear equations with very weak dispersion, which all feature highest waves. Recent work by Ehrnström, Maehlen and Varholm establishes leading-order asymptotics at the crest of solutions to such Whitham-type equations, and conjectures similar expansions for all derivatives of the solution. By extending their techniques we confirm the conjecture for a range of equations with highest waves of Hölder regularity Cs for s∈[0.35,1). We do this by strong induction, redistributing difference operators to deal with singularities in higher-order derivatives arising from integral convolution kernels. The restriction s≥0.35 arises solely from the separate argument establishing the zeroth-order asymptotics, which requires a uniform sign estimate that we verify using rigorous interval arithmetic. The higher-order induction itself applies for every s∈(0,1), so any extension of the zeroth-order result immediately yields the corresponding higher-order expansions.
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BibTeX
@misc{lien2026higherorder,
title = {{Higher-order estimates of highest waves of the Whitham equation}},
author = {Robin Østern Lien},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24447},
eprint = {2609.24447},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24447},
}
RIS
TY - UNPB
TI - Higher-order estimates of highest waves of the Whitham equation
AU - Robin Østern Lien
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24447
UR - https://arxiv.org/abs/2609.24447
AB - The Whitham equation has given its name to a wider family of nonlocal, nonlinear equations with very weak dispersion, which all feature highest waves. Recent work by Ehrnström, Maehlen and Varholm establishes leading-order asymptotics at the crest of solutions to such Whitham-type equations, and conjectures similar expansions for all derivatives of the solution. By extending their techniques we confirm the conjecture for a range of equations with highest waves of Hölder regularity C^s for s∈[0.35,1). We do this by strong induction, redistributing difference operators to deal with singularities in higher-order derivatives arising from integral convolution kernels. The restriction s≥0.35 arises solely from the separate argument establishing the zeroth-order asymptotics, which requires a uniform sign estimate that we verify using rigorous interval arithmetic. The higher-order induction itself applies for every s∈(0,1), so any extension of the zeroth-order result immediately yields the corresponding higher-order expansions.
ER -